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Basic Equations

The LO2 frequency used to track a spectral line at a given frequency Frest centered in the IF band (350 $MHz$) is
\begin{displaymath}
Flo2 = \frac{ Frest \times Doppler + S \times 350 + M \times L \times Eps }
{ M \times H + S }
\end{displaymath} (6)

and, from eq. (5), the image rest frequency at the band center is
\begin{displaymath}
F_{I} = \frac{ (M \times H-S) \times Flo2
- M \times L \times Eps + S \times 350}
{Doppler}
\end{displaymath} (7)

A little algebra follows

$ F_{I} = \frac{ \frac{M \times H-S}{M \times H+S} \times ( Frest \times
Dopple...
...0 + M \times L \times Eps ) - M \times L \times Eps
+ S \times 350}
{Doppler} $

$ F_{I} = \frac{ (M \times H-S) \times ( Frest \times Doppler + S \times
350 + ...
...times (S \times 350 - M
\times L \times Eps) }
{(M \times H+S)\times Doppler} $

$ F_{I} = \frac{M \times H-S}{M \times H+S} \times Frest + \frac{ (M\times
H-S+...
...- M \times H+S)
\times M \times L \times Eps }
{(M \times H+S)\times Doppler} $

$ F_{I} = \frac{M \times H-S}{M \times H+S} \times Frest + \frac{ 2 \times
M \t...
...0 - 2 \times S \times M \times L \times Eps
}
{(M \times H+S) \times Doppler} $

cm and result in

\begin{displaymath}
F_{I} = \frac{M \times H-S}{M \times H+S} \times Frest
+ \...
...H \times 350 - L \times Eps)}
{(M \times H+S) \times Doppler}
\end{displaymath} (8)


This result is not independent of the Doppler effect. Accordingly, it the doppler tracking is not exact for the image frequency. and for 2 different values, $D1$ and $D2$, corresponding to different velocities $V1$ and $V2$, we obtain (assuming no change of $H$)
\begin{displaymath}
\delta F_{I} = \frac{D2-D1}{ D2 \times D1} \times
\frac{ ...
...imes M \times ( H \times 350 - L \times Eps) }
{M \times H+S}
\end{displaymath} (9)

or, assuming $V1$ and $V2 << c$, and with $H >> 1$, $M > 1$, the frequency shift in MHz is
\begin{displaymath}
\delta F_{I} = \frac{\delta V}{c} \times 700
\end{displaymath} (10)

or, in velocity (in km.s$^{-1}$)
\begin{displaymath}
\delta V_{I} = \delta V \frac{700}{F_I}
\end{displaymath} (11)


next up previous contents
Next: Consequences Up: Image Frequency Doppler Tracking Previous: Image Frequency Doppler Tracking   Contents
Gildas manager 2018-06-20